## A mathematical analysis of a model of structured population. I.(English)Zbl 1261.47062

The author considers a kinetic model of structured populations $\frac {\partial f}{\partial t}+\frac {\partial f}{\partial a}+\mu (a,l)f(t,a,l)=P_{\eta }f=\int _{0}^{l}\int _{0}^{a}\eta (a,l,a^{\prime },l^{\prime })f(t,a^{\prime },l^{\prime })\text{d}a^{\prime }\text{d}l^{\prime }$ with initial data in $$L^{1}(\Omega ),$$ where $$\Omega =\left \{ (a,l): 0<a<l, \;0<l<l_{2}\right \}$$ with “boundary” conditions $$f(t,0,.)=K\tilde {f}(t)$$, where $$\tilde {f}(t)\:l\in (0,l_2)\rightarrow f(t,l,l)$$ and the boundary operator $$K$$ is linear and bounded on $$L^1(0,l_{2})$$ with norm $$\left \| K\right \| \geqslant 1$$. The main concern of the paper is the well-posedness of this equation in $$L^{1}$$ spaces in the sense of semigroup theory. Since $$\mu \in L^{\infty }(\Omega )$$ and $$P_{\eta }\in \mathcal {L} (L^{1}(\Omega ))$$, without loss of generality, the author restricts himself to the case $$\mu =0$$ and $$P_{\eta }=0$$. (We point out that generation theory is standard when $$\left \| K\right \| <1$$ for general divergence free vector fields and the resulting semigroups are contractive; see, e.g., [W. Greenberg et al., Boundary value problems in abstract kinetic theory. Basel-Boston-Stuttgart: Birkhäuser Verlag (1987; Zbl 0624.35003)], Chapter 12.)
The author shows that, if $$1_{\varepsilon }$$ denotes the multiplication operator by the indicator function of the interval $$\left ( 0,\varepsilon \right )$$ and if $$\lim _{\varepsilon \rightarrow 0}\left \| K1_{\varepsilon }\right \| _{\mathcal {L}(L^{1}(\Omega ))}<1$$, then the unbounded operator $$- \frac {\partial f}{\partial a}$$ on $$L^{1}(\Omega )$$ with boundary condition $$f(0,.)=K\tilde {f}$$ (where $$\tilde {f}$$ is the trace of $$f$$ on $$\left \{ (l,l): 0<l<l_{2}\right \}$$) generates a positive semigroup. The proof consists in using a change of unknown which transforms (in an equivalent way) the initial problem into another one with boundary operator with norm $$<1$$. This result is already known in all $$L^{p}(\Omega )$$ spaces ($$1\leq p<+\infty$$) with essentially the same proof (see Theorem 3.2 and Theorem 3.3 in [B. Lods and M. Mokhtar-Kharroubi, J. Math. Anal. Appl. 266, No. 1, 70–99 (2002; Zbl 1056.92047)]). Moreover, this result holds for general divergence free vector fields in very general geometries (see Theorem 5.1 in [L. Arlotti et al., Mediterr. J. Math. 8, No. 1, 1–35 (2011; Zbl 1230.47073)]).
To provide concrete generation theorems, the author attempts to show (in his Proposition 3.1) that, if $$K=K_{b}+K_{c}$$, where $$\left \| K_{b}\right \| <1$$ and $$K_{c}$$ is a compact operator, then $\lim _{\varepsilon \rightarrow 0}\left \| K1_{\varepsilon }\right \| _{\mathcal {L}(L^{1}(\Omega ))}<1.$ The generation result behind this is probably true, this is the case in $$L^{p}(\Omega )$$ spaces with $$1<p<+\infty$$ (see Corollary 3.1 in [B. Lods and M. Mokhtar-Kharroubi, J. Math. Anal. Appl. 266, No. 1, 70–99 (2002; Zbl 1056.92047)]), but the proof of the author is, alas, erroneous. Indeed, using the unit ball $$B(0,1)$$ of $$L^{1}(0,l_{2})$$ and the precompactness of $$K_{c}1_{\varepsilon }(B(0,1))$$, for any $$\alpha >0$$ the author recovers that $$K_{c}1_{\varepsilon }(B(0,1))$$ can be covered by a finite number $$n_{\alpha }$$ of balls $$B(K_{c}1_{\varepsilon }(\psi _{l}),\alpha )$$, $$1\leq l\leq n_{\alpha }$$ with radius $$\alpha$$ centered at $$K_{c}1_{\varepsilon }(\psi _{l})$$ with $$\psi _{l}\in B(0,1)$$ and (forgetting that $$n_{\alpha }$$ and the $$\psi _{l}^{\prime }s$$ should depend also on $$\varepsilon$$) passes to the limit as $$\varepsilon \rightarrow 0$$ to arrive at $$\left \| K_{c}1_{\varepsilon }\right \| \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$. Actually, $$K_{c}$$ is a kernel operator, so $$\left \| K_{c}1_{\varepsilon }\right \| =\sup _{l^{\prime }\in \left [ 0,\varepsilon \right ] }\int _{0}^{l_{2}}k(l,l^{\prime })\text{d}l$$ and, in general, $$\lim _{\varepsilon \rightarrow 0}\left \| K_{c}1_{\varepsilon }\right \|$$ need not be equal to zero.

### MSC:

 47D06 One-parameter semigroups and linear evolution equations 92D25 Population dynamics (general)

### Citations:

Zbl 0624.35003; Zbl 1056.92047; Zbl 1230.47073
Full Text: